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294
Mirror symmetry and generalized complex manifolds
, 2004
"... In this paper we develop a relative version of Tduality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n−dimensional smooth real manifold, V a rank n real vector bundle on M, and ∇ a flat connection on V. We define the notion of a ∇−semiflat ge ..."
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Cited by 31 (1 self)
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In this paper we develop a relative version of Tduality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n−dimensional smooth real manifold, V a rank n real vector bundle on M, and ∇ a flat connection on V. We define the notion of a ∇−semiflat generalized complex structure on the total space of V. We show that there is an explicit bijective correspondence between ∇−semiflat generalized complex structures on the total space of V and ∇ ∨ −semiflat generalized complex structures on the total space of V ∨. Similarly we define semiflat generalized complex structures on real n−torus bundles with section over an ndimensional base and establish a similar bijective correspondence between semiflat generalized complex structures on pair of dual torus bundles. Along the way, we give methods of constructing generalized complex structures on the total spaces of vector bundles and torus bundles with sections. We also show that semiflat generalized complex structures give rise to a pair of transverse Dirac structures on the base manifold. We give interpretations of these results in terms of relationships between the cohomology of torus bundles and their duals. We also study the ways in which our results generalize some well established aspects of mirror symmetry as well as some recent proposals relating generalized complex geometry to string theory.
Boyarchenko M., Submanifolds of generalized complex manifolds
 J. Symplectic Geom
"... The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin in [Gua], [H3] (we call these “generalized Lagrangian submanifolds ” in our paper), we introduce and study th ..."
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Cited by 27 (2 self)
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The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin in [Gua], [H3] (we call these “generalized Lagrangian submanifolds ” in our paper), we introduce and study three other classes of submanifolds and their relationships. For generalized complex manifolds that arise from complex (resp., symplectic) manifolds, all three classes specialize to complex (resp., symplectic) submanifolds. In general, however, all three classes are distinct. We discuss some interesting features of our theory of submanifolds, and illustrate them with a few nontrivial examples. Along the way, we obtain a complete and explicit classification of all linear generalized complex structures. We then support our “symplectic/Lagrangian viewpoint” on the submanifolds introduced in [Gua], [H3] by defining the “generalized complex category”, modelled on the constructions of GuilleminSternberg [GS] and Weinstein [W2]. We argue that our approach may be useful for the quantization of generalized complex manifolds. Contents
Reduction and submanifolds of generalized complex manifolds
, 2005
"... We recall the presentation of the generalized, complex structures by classical tensor fields, while noticing that one has a similar presentation and the same integrability conditions for generalized, paracomplex and subtangent structures. This presentation shows that the generalized, complex, parac ..."
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Cited by 26 (3 self)
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We recall the presentation of the generalized, complex structures by classical tensor fields, while noticing that one has a similar presentation and the same integrability conditions for generalized, paracomplex and subtangent structures. This presentation shows that the generalized, complex, paracomplex and subtangent structures belong to the realm of Poisson geometry. Then, we prove geometric reduction theorems of MarsdenRatiu and MarsdenWeinstein type for the mentioned generalized structures and give the characterization of the submanifolds that inherit an induced structure via the corresponding classical tensor fields.
Hamiltonian symmetries and reduction in generalized geometry
"... Abstract. Given a close 3form H ∈ Ω3 0 (M), we define a twisted bracket on the space Γ(TM) ⊕ Ω2 0 (M). We define the group of Htwisted Hamiltonian symmetries Ham(M, J; H) as well as Hamiltonian action of Lie group and moment map in the category of (twisted) generalized complex manifold, which lea ..."
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Cited by 23 (5 self)
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Abstract. Given a close 3form H ∈ Ω3 0 (M), we define a twisted bracket on the space Γ(TM) ⊕ Ω2 0 (M). We define the group of Htwisted Hamiltonian symmetries Ham(M, J; H) as well as Hamiltonian action of Lie group and moment map in the category of (twisted) generalized complex manifold, which leads to generalized complex reduction much the same way as symplectic reduction is constructed. The definitions and constructions are natural extensions of the corresponding ones in symplectic geometry. We describe cutting in generalized complex geometry to show that a general phenomenon in generalized geometry is that topology change is often accompanied by twisting (class) change. 1.
Symplectic forms and cohomology decomposition of almost complex 4manifolds, preprint arXiv: 0812.3680
"... In this paper we continue to study differential forms on an almost complex 4–manifold (M,J) following [18]. We are particularly interested in the subgroups H + J ..."
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Cited by 20 (7 self)
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In this paper we continue to study differential forms on an almost complex 4–manifold (M,J) following [18]. We are particularly interested in the subgroups H + J
ON GLOBAL DEFORMATION QUANTIZATION IN THE ALGEBRAIC CASE
, 2006
"... We give a proof of Yekutieli’s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems. Instead we use an argument inspired by algebraic De Rham cohomology. ..."
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Cited by 20 (1 self)
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We give a proof of Yekutieli’s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems. Instead we use an argument inspired by algebraic De Rham cohomology.
Supersymmetry of the chiral de Rham complex
, 2006
"... We present a superfield formulation of the chiral de Rham complex (CDR) [MSV99] in the setting of a general smooth manifold, and use it to endow CDR with superconformal structures of geometric origin. Given a Riemannian metric, we construct an N = 1 structure on CDR (action of the N = 1 super–Vira ..."
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Cited by 19 (4 self)
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We present a superfield formulation of the chiral de Rham complex (CDR) [MSV99] in the setting of a general smooth manifold, and use it to endow CDR with superconformal structures of geometric origin. Given a Riemannian metric, we construct an N = 1 structure on CDR (action of the N = 1 super–Virasoro, or Neveu–Schwarz, algebra). If the metric is Kähler, and the manifold Ricciflat, this is augmented to an N = 2 structure. Finally, if the manifold is hyperkähler, we obtain an N = 4 structure. The superconformal structures are constructed directly from the LeviCivita connection. These structures provide an analog for CDR of the extended supersymmetries of nonlinear σ–models.
Reduction of generalized complex structures
"... We study reduction of generalized complex structures. More precisely, we investigate the following question. Let J be a generalized complex structure on a manifold M, which admits an action of a Lie group G preserving J. Assume that M0 is a Ginvariant smooth submanifold and the Gaction on M0 is pr ..."
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Cited by 18 (0 self)
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We study reduction of generalized complex structures. More precisely, we investigate the following question. Let J be a generalized complex structure on a manifold M, which admits an action of a Lie group G preserving J. Assume that M0 is a Ginvariant smooth submanifold and the Gaction on M0 is proper and free so that MG: = M0/G is a smooth manifold. Under what condition does J descend to a generalized complex structure on MG? We describe a sufficient condition for the reduction to hold, which includes the MarsdenWeinstein reduction of symplectic manifolds and the reduction of the complex structures in Kähler manifolds as special cases. As an application, we study reduction of generalized Kähler manifolds.
The Lefschetz property, formality and blowing up in symplectic geometry
, 2005
"... In this paper we study the behaviour of the Lefschetz property under the blowup construction. We show that it is possible to reduce the dimension of the kernel of the Lefschetz map if we blow up along a suitable submanifold satisfying the Lefschetz property. We use that, together with results abo ..."
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Cited by 18 (4 self)
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In this paper we study the behaviour of the Lefschetz property under the blowup construction. We show that it is possible to reduce the dimension of the kernel of the Lefschetz map if we blow up along a suitable submanifold satisfying the Lefschetz property. We use that, together with results about Massey products, to construct compact nonformal symplectic manifolds satisfying the Lefschetz property.